Tag: understanding 3d and 2d shapes

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Questions Related to understanding 3d and 2d shapes

Multiple choice maths understanding 3d and 2d shapes defining regular polygons sum of exterior angles of a polygon regular polygons

The area of a regular polygon of $2n$ sides inscribed in a circle is given by?

  1. The geometric mean of the areas of the inscribed and circumscribed polygons of $n$ sides.
  2. The arithmetic mean of the areas of the inscribed and circumscribed polygons of $n$ sides.
  3. The harmonic mean of the areas of the inscribed and circumscribed polygons of $n$ sides.

  4. None of the above
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Let $a$ be the radius of the circle 


Then,$\displaystyle s _{1}= $ Area of regular polygon of n sides inscribed in the circle $\displaystyle =\frac{1}{2}na^{2}\sin\left ( \frac{2\pi }{n} \right )$

$\displaystyle s _{2}= $  Area of regular polygon of n sides circumscribing in the circle $\displaystyle  = na^{2}\tan \frac{\pi }{n}$

$\displaystyle s _{3}= $ Area of regular polygon of 2n sides inscribed in the circle $\displaystyle  = na^{2}\tan \frac{\pi }{n}$ 

[replacing $n$ by $2n$ is $\displaystyle {(S _{1}}$]

$\displaystyle \therefore $ Geometric mean of $\displaystyle {S _{1}}$ and 

$\displaystyle {S _{2}}$ $\displaystyle = \sqrt{(S _{1}S _{2})}= na^{2}\sin\left ( \frac{\pi }{n}\right ) = S _{3}$

Multiple choice maths understanding 3d and 2d shapes defining regular polygons sum of exterior angles of a polygon regular polygons

If A B C D E F is a regular hexagon with A B = a and B C = b, then CE equals

  1. b-a

  2. -b

  3. b-2a

  4. None of these
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

In a regular hexagon ABCDEF, the vector CE can be found using vector addition. Since AB = a and BC = b, the vectors for the sides are related by the geometry of the hexagon, leading to CE = b - a.

Multiple choice maths understanding 3d and 2d shapes defining regular polygons sum of exterior angles of a polygon regular polygons

Relation between circumradius and number of sides is given by-

  1. $Area=\dfrac{r^2n\sin(\dfrac{360}{n})}{3}$
  2. $Area=\dfrac{r^2n\sin(\dfrac{360}{n})}{2}$
  3. $Area=\dfrac{r^2n\cos(\dfrac{360}{n})}{2}$

  4. None of the above
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The area of a regular polygon with n sides and circumradius r is given by n * (1/2 * r^2 * sin(360/n)). This formula is derived by summing the areas of n isosceles triangles with sides r and included angle 360/n.

Multiple choice maths understanding 3d and 2d shapes defining regular polygons sum of exterior angles of a polygon regular polygons

The sum of the radii of inscribed and circumscribed circles of an n sided regular polygon of side $'a'$ is

  1. $=\frac{a}{2} \left ( \frac{1}{\sin \pi/2x} + \cot \frac{\pi}{x} \right )$
  2. $=\frac{a}{2} \left ( \frac{1}{\sin \pi/x} + \cot \frac{\pi}{2x} \right )$
  3. $=\frac{a}{2} \left ( \frac{1}{\sin \pi/x} + \cot \frac{\pi}{x} \right )$

  4. None of these
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

$R\sin \theta  = \frac{a}{2}$
$R = \frac{a}{2\sin \theta }$
$\tan \theta = \frac{a/2}{r}$
$r = \frac{a}{2\tan \theta }                                   \theta = \frac{2\pi}{n} \times\frac{1}{2}$
$R+r = \frac{a}{2} \left ( \frac{1}{\sin \theta}+\frac{\sin \theta}{\cos \theta } \right )             = \frac{\pi}{x}$
    $= \frac{a}{2} \left ( \frac{1}{\sin \pi/x} + \cot \frac{\pi}{x} \right )$

Multiple choice maths understanding 3d and 2d shapes defining regular polygons sum of exterior angles of a polygon regular polygons

In $\Delta ABC$, there are 35 lines drawn parallel to the base BC such that each line divides the other side into, equal parts. 
If BC =1.8 m find the length of $P _7 Q _7$.

  1. 1.8 m

  2. 3.5 m

  3. 0.35 m

  4. 0.18 m
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Using the properties of parallel lines in a triangle, the length of the segments follows an arithmetic progression. With 35 lines, the segments divide the side into 36 equal parts; the 7th line corresponds to a ratio of 7/36 of the base, but the calculation 1.8 * (7/36) = 0.35 m is correct.